Flexible Body Input File

OAFormulation

The total displacement field

μ

for every point of a flexible body is obtained from the displacement of a local frame defining the rigid motion of the body and from an additional local displacement field

w_{L}

corresponding to the small vibrations of the body.

Figure 1.

( $G_{0}$ , $G_{1}$ , $G_{2}$ , and $G_{3}$ ) defines the global frame ( $e_{1}$ , $e_{2}$ , and $e_{3}$ ).

( $L_{0}$ , $L_{1}$ , $L_{2}$ , and $L_{3}$ ) defines an orthonormal local frame.

$P$ is the rotation matrix from ( $G_{0}$ , $G_{1}$ , $G_{2}$ , and $G_{3}$ ) to ( $L_{0}$ , $L_{1}$ , $L_{2}$ , and $L_{3}$ ).

The total displacement, $u$ , can thus be expressed as:

u = X \cdot u_{L_{1}} + Y \cdot u_{L_{2}} + Z \cdot u_{L_{3}} + (1 - X - Y - Z) \cdot u_{L_{0}} + P w_{L} = u_{R} + P w_{L}

Where, $u_{L_{0}}$ , $u_{L_{1}}$ , $u_{L_{2}}$ , and $u_{L_{3}}$ are displacements of points $L_{0}$ , $L_{1}$ , $L_{2}$ , and $L_{3}$ , respectively,

$X$ , $Y$ , and $Z$ are coordinates in the local frame ( $L_{0}$ , $L_{1}$ , $L_{2}$ , and $L_{3}$ )

$u_{R}$ is the rigid body contribution to the total displacement

Local displacement is given by a combination of local vibration modes $Φ_{L}^{i}$ :

w_{L} = Φ_{L} α

Where, $α$ is the vector of local modal contributions.

Rigid body displacement $u_{R}$ can also be expressed as a combination of 12 modes:

u_{R} = Φ_{R} \cdot {(u_{L_{1}}^{1}, u_{L_{1}}^{2}, u_{L_{1}}^{3}, u_{L_{2}}^{1}, u_{L_{2}}^{2}, u_{L_{2}}^{3}, u_{L_{3}}^{1}, u_{L_{3}}^{2}, u_{L_{3}}^{3}, u_{L_{0}}^{1}, u_{L_{0}}^{2}, u_{L_{0}}^{3})}^{T}

Where the projection modes $Φ_{R}^{i}$ are obtained from the local coordinates:

\begin{matrix} Φ_{R}^{1} = X \cdot e_{1} & Φ_{R}^{2} = X \cdot e_{2} & Φ_{R}^{3} = X \cdot e_{3} \\ Φ_{R}^{4} = Y \cdot e_{1} & Φ_{R}^{5} = Y \cdot e_{2} & Φ_{R}^{6} = Y \cdot e_{3} \\ Φ_{R}^{7} = Z \cdot e_{1} & Φ_{R}^{8} = Z \cdot e_{2} & Φ_{R}^{9} = Z \cdot e_{3} \\ Φ_{R}^{10} = (1 - X - Y - Z) \cdot e_{1} & Φ_{R}^{11} = (1 - X - Y - Z) \cdot e_{2} & Φ_{R}^{12} = (1 - X - Y - Z) \cdot e_{3} \end{matrix}

The choice of the local frame ( $L_{0}$ , $L_{1}$ , $L_{2}$ , and $L_{3}$ ) is fully arbitrary. These points do not need to be input explicitly. Their locations define local coordinates and thus, the components of the modes $Φ_{R}^{i}$ .

If the flexible body contains elements with rotational DOF, three additional modes must be added to the $Φ_{R}^{i}$ family, accounting for the inertia associated with these DOF. The components of these additional modes on each node of the flexible body having rotational DOF are:

Φ_{R}^{13} = [\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{matrix}] Φ_{R}^{14} = [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 1 \\ 0 \end{matrix}] Φ_{R}^{15} = [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 1 \end{matrix}]